Consider $f$ a $C^1$ function. Let $\{x^k\}$ be a sequence such that
$$d^k = f'(x^k) + \epsilon^k, \quad \Vert \epsilon^k \Vert \leq \delta;$$ $$x^{k+1} = x^k - \alpha_k d^k.$$
Show that for every $\delta'> \delta$ there exist $\overline{\alpha} \geq \underline{\alpha} > 0$ such that if $\alpha_k \in [\underline{\alpha}, \overline{\alpha}]$ then $\{x^k\}$ admits a accumulation point $\overline{x}$ such that $\Vert f'(\overline{x}) \Vert \leq \delta'$.
I could only show this using that $f$ is a Lipschitz function. This problem can be found In Bersetkas's book on nonlinear programing and there is a solution in Bersetka's site. However, he also assume that $f$ is Lipschitz. My question is: Is this hypothesis necessary?